Angle Sums

• ## Angle Sums

Examine the angles in a triangle, quadrilateral, pentagon, hexagon, heptagon or octagon. Can you find a relationship between the number of sides and the sum of the interior angles?

### Instructions

• Choose a polygon, and reshape it by dragging the vertices to new locations. As the figure changes shape, the angle measures will automatically update.
• Animation: For triangles and quadrilaterals, you can play an animated clip by clicking the image in the lower right corner. This movie will provide a visual proof for the value of the angle sum.

### Exploration

Begin by exploring triangles and quadrilaterals. As you drag the vertices to reshape these polygons, occasionally play the animated clip in the lower right corner.
• Is there a different result for different shapes?
• Test your conjecture by creating a square, rhombus, parallelogram, and other quadrilaterals. For each type of shape, play the movie. How does the result change?
Use the applet to examine pentagons, hexagons, heptagons and octagons, too. Do you notice a pattern? How does the sum of the angles change as the number of sides changes?
• What happens to the angle sum as you reshape the polygon by moving the original vertices?
• Find a formula that relates the number of sides (n) to the sum of the interior angle measures?

### Objectives and Standards

NCTM Standards and Expectations
• Geometry / Measurement
• Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• 6-8
• Geometry